434 XV NORMED ALGEBRAS AND SPECTRAL THEORY
0. Now, by the definition of the adjoint (15.12.3), we see first that z e dom(#?)
and next that (x | H* z + iz) = 0 for all x e doml/^), and since domC^) is
dense in E, we have H^ - z + iz = 0. Since dom(H) a dom^), we have
dom(#*) <=: dom(#*), and H* is the restriction of #*. Hence H* z + iz = 0,
in other words z e E# . But E# c dom^) by definition, so that
0 = (z | (#* + iJ) - z) = ((# ! - il) z | z) = ((H* - i/) z | z) = -~2i||z||2,
and therefore finally z = 0.
To prove that F = E for all X such that */A > 0, we observe that if this is
true for 1 = A0, then the operator (H^ ^o/)""1 is everywhere defined and
continuous, and satisfies
by virtue of (15.13.10.1). It follows (15.12.11) that (Ht - A/)"1 is everywhere
defined and continuous for |A /10| < |c/A0|. By induction, (H^ A/)"1 is
everywhere defined and continuous for |/l (f)n/| < (f)w, and since the
integer n is arbitrary, we have shown that (Hl /I/)""1 is everywhere defined
for all A in the half-plane JX > 0.
We have thus defined, for all A e C such that Ğ/A > 0, a continuous operator
A(X) = (^ - IT)"1 on Eğ with image equal to dom^) c dom(/P). Since
(H^ /l/)^(l) = /, it follows that, for any two complex numbers a, ft such
that */a > 0 and ,//? > 0, the operator
*(a, 0) = (^ - a/^O?) Ğ / + 08 - aX/0
is everywhere defined and continuous; also the operator A(p)(Hi /?/) is the
identity on dom^), and therefore
(15.13.10.2) K(oc9 0)K(p, y) = K(a, 7), K(a, a) == /.
Hence the operators K(a, fi) are invertible and bicontinuous.
This being so, for every x e dom(H*) we have K(i, X) x e dom(H *) and
(H* - II) - (K(i9 X)-x) = (H* - A/) - jc + (A - i
and likewise (J^T* - //) (JC(A, 0 x) = (/?* - A/) x. This shows that K(i, A)
is a bicontinuous bijection of EjJ onto E+(A). Similarly for E~(A). we